Optimization of energy flow in port - Controlled hamiltonian system

. Chapter 2. Controlling the power flow in the isolator. Chapter 3. Controlling the power flow in the quarter car model Chapter 4. Controlling the power flow in the tuned mass damper. General conclusions. Present the main results obtained in the thesis, new points and next major research directions. 3 CHAPTER 1. OVERVIEW This chapter classifies the control methods based on energy and limits the scope of the methods used in the thesis. The basic systems of the PCH system (Port Controlled Hamilton System) were introduced to describe the dynamics systems and the energy flow equations in these systems. 1.1. Vibration control The control methods base on energy include passive, active and semi-active control. In fact, the control of variation depends on the nature, characteristics and also on the cost (due to the active control with high cost) of each specific project. This thesis studies only passive and semi-active control. 1.2. Energy flow analysis The variable studied in energy flow analysis is a combination of effects due to force, velocity, and their product, power, ie the rate of energy change. This combination plays a role as a unique parameter to describe the dynamics and responses of a system, including and reflecting fully the equilibrium and motion of the system. Therefore overcome limitations in the study of separate displacement response and force response. 1.3. PCH (Port- Controlled Hamiltonian Systems) The classic Hamilton system can be rewritten in the following form ( ) H z H z  = +   =  T z J - R Gu y G (1.5) where z is the system state, H is the Hamiltonian function, u is the input of the system, G is the input distribution matrix, J satisfying 4 T J = -J is the lossless interaction structure matrix, R satisfying T R = -R , is the dissipation structure matrix, y is the output of the system. The equation system (1.9) called the PCH system was first introduced by Maschke and Van der Schaft (1999). The equation "energy flow" has the form T H H H z z    = −      T u y R (1.9) where the left side is the change of Hamiltonian (which is usually chosen as a total of kinetic and potential energy). The first term in the right side is the "energy flow" or “power flow” into the system characterized by power variables u and y. The second term is the energy flow dissipated through the matrix R. 1.4. Issues of the thesis 1.4.1. Research situation The domestic research on vibration control has been carried out a lot, but most studies are based on the classical approach, not considering the energy flow aspect. The studies of semi-active control were also initially studied domestically and applied to brake, clutch and force feedback devices. However, the control laws are still based on the classic approach, and at the same time, the law of semi-active control has not been shown. 1.4.2. The problem of the thesis The thesis focuses on the analytical studies of optimizing parameters in passive control or proposing semi active control algorithms based on the energy flow applied to three specific types of vibration control systems, with increasing degrees of freedom, from simple to complex. 5 CHAPTER 2. CONTROLLING THE ENERGY FLOW IN THE ISOLATOR This chapter considers the problem of vibration control using the single degree of fredom isolator. After making the energy flow formulas in the isolator, the thesis will study the effect of the parameters of the isolator in the passive case with the assumption of harmonic motion. In the case of semi-active, the thesis proposes an on / off control algorithm based on the energy flow and corrects this algorithm based on the optimal control algorithm. 2.1. Concept of vibration isolator Vibration isolation, essentially, involves inserting an elastic (or isolation) component between the vibration mass and the source of vibration to reduce the system dynamic response. Vibration isolation can be achieved by passive, semi-active and active modes. 2.2. The energy flow in the vibration isolator Consider the details of an isolator as shown in Figure 2.5. k c m r x Figure 2.5: Vibration isolator model The dimensionless motion equation has the form PCH: ( ) 0 1 21 0 rx r x r r xx x  −− −        = +        −−        (2.9) 6 In which ( )( )22 0 11 ; , , 1 02 x r H x x r x −    = + − = =   −    z J ( ) 1 0 , 0; ; 2 0 1 r r x −    = = =   −    u R G y = z (2.10) Energy flow into the system: ( ) ( )2P r x r r x x= − − + − (2.11) where 2 c km  = is the damping ratio of the isolator The energy flow (2.11) consists of two components: the energy flowing from the base into the spring ( )1P r x r= − − and the energy flowing from the damper into the isolated mass ( )2 2P r x x= − . Because damping is a controlled quantity, we consider direct control of the P2 energy flow component. 2.3. Effect of the isolator parameters In the case of hamonic external foundation 0 cosr r = , 1 n    = is the dimensionless frequency, nt = is the dimensionless time, applying the analytical methods, we obtain the energy flow P2 consisting of three terms, two terms oscillating with the frequency 2 (called the oscillating energy flow) and the third term constant (called the average energy flow). 7 ( ) ( )( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 22 4 0 2 2 2 4 4 2 22 2 2 2 2 2 2 cos2 1 4 1 4 1 4 2 sin 2 1 1 1 4 8 4 r P                       − +     − +−  =   − +   − +   − + + − +   (2.23) Solving the min-max optimization problem for the average energy flow ( )min max tbP   , we obtain the optimal value: 0 1 3  = (2.37) is the damping ratio for the isolator to take energy out of the system the most in the case of passive control. 2.4. On off damping controller based on the energy flow Considering the semi-active control problem, the damping ratio can vary between h and l. Based on the energy flow formula (2.11), consider the control law: ( )  ( )  2 2 sgn sgn 0 sgn sgn 0 h l P r x x P r x x     = −  =  = −  (2.38) If P2<0 means that the damper is taking power out of the system, the damping is set to the on value of h to activate the damper. Conversely, if the dampers are putting energy into the system (P2>0), set the damping value at l to limit the damper operation. We perform numerical simulations in 3 cases of excitation frequency: resonance, fixed point and above fixed point, corresponding to non- 8 dimensional frequencies of 1, 2 and 2 to illustrate the effectiveness of the semi active control algorithm. In three cases, the on off damping controller is the best. In the case of fixed points (figure 2.7), large and small damping create oscillation with a non-dimensional amplitude equal to 1. However, on off controller create smaller oscillation. This means that the on off damping has overcome the inherent limitations of passive damping. Fig 2.6: Amplitude case =1 Fig 2.7: Amplitude case = 2 Fig 2.8: Amplitude case =2 2.5. Adjustment based on controlling the optimal on off damping 9 In this section, the thesis presents an on off control law for the best effect in class. Control rules based on the energy flow (2.38) (which is a separate case in the class of control laws considered) can be adjusted according to the optimal control law presented. Using time-shifting technique and harmonic balance method, in case of harmonic oscillation, find the lower bound: ( )( ) ( ) ( ) 2 3 0 2 2 2 2 2 2 0 0 2 2 0 2 2 2 2 2 sin cos2 4 sin 1 min 1 2 sin sin 2 4 sin h l l opt h l h l l L t h l opt h l h l l t t t t t t J r t t t t t                                  −  − + +    +   − −    + − + −            = −  − −     − −    + − +          ( ) 2 2 2 1 1        +   −     (2.56) LJ is the smallest amplitude response in the class of all the optimal control rules whose time shifting depends on the product of any two harmonic functions. The thesis proposes an improved form of controller (2.38) based on the lower bound response (2.56): ( ) ( )  ( ) ( )  2 2 sgn 0 sgn 0 h l r x x r x r x x r x       − + −  =  − + −  (2.57) Where  is a adjustment parameter, changed so that the controller (2.57) is tracing to (2.56). 10 Table 2.1. The amplitude of x varies with the adjustment parameter  = - 0 (not improved) 0.5 1  Amplitude of x/r0 1 0.7 0.6 0.7 1 the lower bound JL/r0 at = 2 0.6 The results in Table 2.1 show that the adjustment value  = 0.5 gives slightly better efficiency than the case without correction  = 0. 2.6. Conclusion of chapter 2 In the case of passive control, with the assumption of foundation in harmonic form, the thesis shows the optimal damping ratio value of the isolator In the case of semi-active control, the thesis has proposed an on off controller based on the energy flow and perform numerical simulations in three cases of excitation frequency: resonance, fixed point and above fixed point, corresponding to the dimensionless frequency equal to 1, 2 and 2. The calculated results all conclude that the on / off resistance is much more effective than the large passive damping and small passive damping. In order to improve the efficiency, the thesis will export the controller by a parameter  based on the lower bound response (which is the response of the optimal on / off control law). The results show that the calibration parameter is about 0.5 for the response at the frequency of the fixed point (ω = 2 ) to reach the theoretical optimal value. The results of this chapter are presented in the articles [T1], [T2], [T7] 11 CHAPTER 3. CONTROLLING ENERGY FLOW IN A QUARTER CAR MODEL Similar to the procedure in the previous chapter, this chapter considers the oscillation control problem for a typical 2-degree-order mechanical system, which is a quarter car model. 3.1. Concept of car suspension Conventional suspension is made up of 3 main parts: elastic component, shock-absorbers and a set of mechanical components remaining. The quarter car model describes the interaction between the suspension, the tire and the body at ¼ the vehicle. Instead of using classic criteria for comfort and road-holding, it is possible to use an energy flow criteria to unify the design of both problems. 3.2. Power flow formulations Consider the quarter car model in Figure 3.3. Fig 3.3. the quarter car model Similar to the previous chapter, through the Hamilton function, it is possible to write motion equations in the form of PCH. In the case of the road profile is presented in the harmonic forrm, using the analytical methods, the average energy flow as: ( ) ( )( ) 6 2 2 2 0 2 2 4 2 2 2 2 2 2 t s tb s t t s t t s t s t K x bM P M M K M M K K K M K M M b       =   − + +      + −     + − +  (3.24) 12 3.3. Effect of shock absorber damping Solve the min-max optimization problem for the average energy power ( )min max tb b P  We have the optimal solution of damping: ( ) ( ) 4 2 0 0 0 2 0 0 s t s t s t t t s t M M M K KM KM KK b K M M    − = − + + + + (3.33) Consider a numerical example of a motorcycle as follows: Ms=117kg, Mt=30kg, K=26000 N/m, Kt=200000 N/m, x0=1cm. 0 2 1 20Tần số (Hz) P tb ( 1 0 3 W ) b=1000Ns/m b=b0 b=10000Ns/m Figure 3.4: Average energy flow with different absorber’s damping The results show that both too large and too small damping increas the flow of energy into the system. 3.4. On off damping controller based on the energy flow Considering the semi-active control problem with the aim of the vehicle's comfort, the energy flow into the system has the form: ( )st st sP Kx bx x= − − (3.41) The proposed controller based on the energy flow formula (3.40) has the form:     sgn 0 sgn 0 h st s l st s b x x b b x x   =   (3.42) 13 If the absorber is taking power out of the body mass, the damping is set to bh to enable the absorber. Conversely, set the damping value at bl to limit the damping operation. With the data given above, in addition consider bl = 700Ns / m, bh = 3000Ns / m. The calculation results clearly show the effectiveness of the on / off controller based on the energy flow when compared to the case of passive control. Figure 3.7: Frequency response of vibration amplitude of body mass 3.5. Adjustment based on controlling the optimal on off damping Similar to section 2.5, this section presents an improved form of controller (3.41) based on the lower bound. However, the difference here is that this controller for quarter car model is first found by us and published it in the article [T2]. In case of harmonic external force: ( )0 cosf f t = − (3.49) Using time-shifting techniques and harmonic balancing methods, performance index was found ( ) ( ) 2 2 0 2 1 3 4cos sin sin cosAJ f a a a a   = + + + (3.60) With ( ) ( ) ( ) ( )( ) 1 1 1 T ss e sc ss e sca b b b b b b  − − = − − + − − + − + + fr I D A D D I D A D D H 14 ( ) ( ) ( ) ( )( ) 1 1 2 T ss e sc ss e sca b b b b b b  − − = − + − − − − + + fr I D A D D I D A D D H ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) 1 1 3 1 1 4 T ss e sc ss e sc T ss e sc ss e sc a b b b b b b a b b b b b b     − − − − = − + + + − + + − = − + + + + + − f f r I D A D D I D A D D H r I D A D D I D A D D H 2, sin , sin cosh l h l h le l s ss s sc s s b b b b b b b b t b t b t t       − − − = + = = (3.62) And lower bound amplitude ( ) ( )( ) ( ) ( )( )2 2 2 22 2 2 21 2 3 4 1 4 2 3 1 4 2 3 0 0 min 2 s L t J a a a a a a a a a a a a f     = + + + − − + + + + − = (3.66) Where A is the system matrix, D is the location matrix of a single on / off damper with on off damping b, Hf is the input location vector, bh and bl respectively are the on-value and off-value, vector r expresses the location of the target states. Considering the adjusment controller of (3.42) has the form:     2 2 sgn 0 sgn 0 h st s st l st s st b x x x b b x x x    +  =  +  (3.67) Where  is a improved coefficient found by solving the minimum problem: min JE  (3.68) Với: ( ) ( ) i J A i L iE J J   = − (3.69) Where i is the tracking frequency, JA is the performance index of the designed controller and JL is the lower bound (3.66). 15 Figure 3.8: Frequency response of vibration amplitude of body mass The results in Figure 3.8 show that the curve produced by the improved controller nearly coincides with that of the lower bound controller. 3.6. Conclusion of chapter 3 In the case of passive vibration control, the thesis has shown the optimal analytic solution of the shock absorbers based on the energy flow cretaria. In the case of semi-proactive, the thesis has proposed an on off controller based on the energy flow. Through numerical simulations in specific cases, the calculation results clearly show the efficiency of the controller when compared with the case of passive control. To consider this real adjustment, the thesis has found an optimal on off controller in the controller class with the time shifting depending on the product of the 2 harmonic functions, applied to a quarter car model. From the response of the lower bound amplitude of the optimal controller, the thesis found the scalar coefficient and shown the frequency response that closely matches the lower bound curve. The chapter results are presented in the articles [T2], [T4], [T6]. 16 CHAPTER 4. CONTROLLING ENERGY FLOW IN TUNED MASS DAMPER This chapter studies the oscillation control problem for the model of mass dampers of one and many degrees of freedom. In the semi- active case, this chapter proposes several versions of the on / off control controller based on the energy flow and improved this controller. 4.1. Concept of tuned mass dampers Tuned mass dampers (TMDs) are one (or more) of the auxiliary mass installed into the main structure via connections, typically springs and dampers. TMD systems with the on / off damping do not have an accurate solution. Most of the studies in the literature use numerical methods. The approximate analytic solutions on many degrees of degrees of freedom are hardly seen and this is the research goal of this chapter. 4.2. Power flow formulations Consider a set of TMD as shown in Figure 4.10. Figure 4.10: Model of TMD installation system With: , , , , , 2 d d d d s d d d d s s m k ck m m m m            = = = = = = (4.6) The maximum value of the energy flow is the sum of the amplitudes of oscillating energy with the average energy. The external dimensionless maximum energy flow Pm from the outside into the 17 whole system (including main system and TMD system) has the form: ( ) ( )( ) ( ) ( )( ) ( ) 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 5 2 2 4 2 2 2 2 2 2 2 2 2 4 1 4 1 1 4 1 mP                               − + = +  − + + −     + + −  +  − + + −     + + −  (4.24) The non-dimensional external maximum energy flow Pm from outside plus the TMD word transmitted to the main system has the following form: ( ) ( )( ) ( ) 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 4 1 1 4 1 mP                   − + = −  − + + −     + + −  (4.29) 4.3. Effect of parameters of tuned mass damper Solve the min-max optimization problem for maximum energy flow ( ) , min max mP    in 2 cases (4.24) or (4.29). The optimal solutions opt and opt found using this numerical method are compared to the Den Hartog solution: ( )1 1 1 3 , 1 8 1      = = + + (4.30) The comparison results show that in the case of energy flow into the system, the optimal solution is very close to that of Den Hartog. 18 However, in the case of an external power flow into the main system, the optimal parameters differ significantly from those of Den Hartog. This shows that the solution of Den Hartog is not large enough to limit the flow of energy into the main system. Figure 4.11: Maximum energy flowing into the system with =5% 1 10.9524, 0.1303; 0.9561, 0.1336opt opt     = = = = Figure 4.13: Maximum energy flowing into the main system with =5% 1 10.9524, 0.1303; 0.91, 0.20opt opt     = = = = 4.4. On and off damping controller based on the energy flow ma ka ca s v r TMD Một phần của hệ chịu kích động ngoài Một phần của hệ gắn với TMD Một phần của hệ cần được kiểm soát dao động Figure 4.15: The MDOF system attached with dynamic vibration absober Denote the location of the single harmonic loading by the vector s, the loccation of the target mass by the vector r, and the location of the TMD by the vector v. The symbols ma, ka and ca are respectively the mass, the stiffness, and the on / off damping of TMD. 4.4.1. Controller for maximal flow of energy to TMD - version 1 19 In this controller’s version, it is aimed to maximize the vibrational energy of the absorber to attract the energy from the MDOF main system. The power-driven controller (version 1) is proposed as follows:     sgn 0 sgn 0 T h a a a T l a a c m x c c m x   =   v x v x (4.41) Meaning that if the power flow is into the TMD, the small damping value cl is used to activate the TMD. On the contrary, with the undesirable trend, the large damping value ch is only used to restrict TMD activities. 4.4.2. Controller of minimum energy flow into the system - version 2 In this controller’s version, it is aimed to minimize the energy injecting to the whole system. The power-driven controller as follows: 0 0 T h a T l c f c c f   =   x s x s (4.46) This means that if energy flows into the system, a large damping value ch is used to limit TMD activity. In contrast, the small damping value cl is used to activate TMD. 4.5. Adjustment based on the optimal on off daming Similar to sections 2.5 and 3.5, in this section, the thesis proposes the lower bound response generated by the optimal on off controller. This is a new result of the thesis and was presented in [T3]. The exitation is assumed as harmonic form ( )0 cosf f t = + , the 20 undamped transfer function HAB between two certain vectors a and b and is defined as: ( )( ) 1 2T T AB BA aH H m − = = − +a K M vv b (4.56) Using time-shifting technique, harmonic balance method, through many steps of transformation, we can find the performance index: ( )2 4 22 2 2 2 2 0 2 1 2 3 1 3 2 2 2 1 1 2 3 2 3 A s RV VS RS a RS e J t H H H m H f a a a a a a T c a a a a a a   = + − + −  + + −  (4.70) With ( )4 21 2 4 2 2 2 2 2 2 2 3 2 sin 2 sin e a RV VS RS h l a RV VS RS s e h l RS e s a c m H H H T c c a m H H T H T t c c c a H c t          = +   −   = + − +        − = 2 4 2 2 , , sin , sin cos h l e l s a a a VV h l h l ss s sc s s c c c c t T k m m H c c c c c t c t t          − = + = − − − − = = (4.61) The lower bound JL is obtained by minimizing the function (4.70) with the single parameter ts. Based on this lower bound solution, the power driven controller version 1 (4.41) is improved as: 2 2 0 0 T h a a h a a T l a a h a c m x c x c c m x c x    +  =  +  v x v x (4.71) And the power driven controller version 2 (4.46) is improved as: 2 2 0 0 T h h a a T l h a c f c x c c f c x    −  =  −  x s x s (4.72) 21 where  is a constant parameter. The parameters  can be optimized to make the controller trace as close as posible to the optimal bound controller. 4.6. Example of numerical calculations ma ka ca k1 m4 m1 m3 m2 f1 f3 f2 f4 k2 k3 k4 k5 k6 k7 Figure 4.16: The 4 DOF system attached with on / off damping absorber Table 4.1. Numerical values of the main system’s parameters index 1 2 3 4 5 6 7 k (N/m) 30,000 30,000 20,000 50,000 20,000 30,000 45,000 m (kg) 4 10 4 8 F (N) 1 1 1 1 Naturel frequency (Hz) 8.196 12.250 22.621 33.279 Table 4.2. Passive absorber’s parameters of TMD, tuned to first mode ma=0.5kg TMD attached to mass #1 mass #2 mass #3 mass #4 ka (N/m) 1301.1 1225.3 1315.2 1375.9 Passive damping cp (Ns/m) 3.7 7.0 3.0 2.3 22 Table 4.3. The ratio of on damping and off damping to passive damping in the case study Passive ch=cl=cp On off ch=2cp, cl=0.2cp Table 4.4: Parameters of power driven controller TMD attached to mass #1 mass #2 mass #3 mass #4 Traced frequency (Hz) 7.64; 8.12; 8.54 7.22; 7.82; 8.78 7.82; 8.12; 8.48 7.94; 8.18; 8.42 parameter  (controller version 1) 0.1 0.1 0.1 0.0 parameter  (controller version 2) 0.4 0.2 0.4 0.6 Figure 4.17: Frequency response when TMD attaches to mass # 1; Figure 4.18: Frequency response when TMD attaches to mass #2; 23 Figure 4.19: Frequency response when TMD attaches to mass #3; Figure 4.20: Frequency response when TMD attaches to mass # 4; Some observations are drawn: - Both controller versions work well and give better results than passive control. - Calculating at the frequency of 3 inflection points greatly reduces parameters optimization. 4.7. Conclusion of chapter 4 In the passive case, the optimal parameters based on the flow of energy from outside to the system and the flow of energy from outside plus the flow of energy from TMD into the main system have been found by numerical methods and compared to the Den Hartog solution, which applies to 1 DOF system. In the case of semi-active, the thesis has proposed two versions of the on off damping controller based on two different energy flow criteritions, applicable to multi degrees of freedom system attached to TMD. In order to improve these two versions better, the thesis created a lower bound response, numerical simulation calculations were performed to clarify th